Devices and techniques for logical processing

ABSTRACT

The invention is directed to apparatus, methods, systems and computer programs which permit a simplification or reduction of the description of a complex digital circuit. This is accomplished by using a completely new system of propositional logic ( 13 B), representing the logic of a logical circuit to be designed as points and vectors in a vector space, simplifying the logic so represented to a simpler form using vector transformations, and designing the circuit using the simpler form. The invention is also directed to apparatus, methods, systems and devices which will implement the vector logic system in optical forms, including free-space optical methods, flat-optical methods, colorimetric methods, and methods using a polarization-based AND-gate.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims priority from and is related to U.S. provisionalapplication Ser. No. 60/238,007, filed Oct. 6, 2000, entitled: OPTICALAND GATE AND SWITCHING DEVICE USING POLARIZATION PHOTOCHROMISM. Thecontents of that provisional are hereby incorporated by reference intheir entirety.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates to the field of logical processing and, moreparticularly to devices and techniques for implementing and forsimplifying digital logic.

The invention relates to the field of logical processing, to devices andtechniques for simplifying two- and multi-level digital logic, using theproposed vector form of the digital logic, and to devices and techniquesfor implementing the said vector logic in the form of opticalprocessors, circuits composed of optical logic gates or switches,including therefore both optical multiplexers and opticaldemultiplexers, as well as integrated optical circuits.

2. Description of Related Art

Logic can be described as techniques and operations by which one movesfrom what one knows to be true to new truths. The principles of logichave been applied in the design and operation of digital logic circuits.Modern-day computers and other processing devices have utilized digitallogic extensively. Many of the problems to which digital logic can beapplied are complex, involving many independent variables. This resultsin extremely complex logical circuits in which large numbers ofoperations are performed. The cost associated with manufacturing andfabrication of such complex digital circuits is great. It would behighly desirable to reduce the size of these circuits while preservingthe same functionality, and thereby to reduce the manufacturing cost oftheir fabrication and to improve their performance and speed.

A number of optically active materials are known in the art. Among theseare photochromic polymers as described in an article by KunihuroIchimura, “Photochromic Polymers”, in a text by John C. Crano and RobertJ. Gugilelmetti, eds., entitled Organic Photochromic and ThermochromicCompounds, Vol. 2, Kluwer, New York, 1999, pp. 9-63, the contents ofwhich are incorporated herein by reference thereto.

Digital computers are of course well known, but more recently opticalcomputers have been developed which can perform logical functions usingoptical elements. These logical processors can in principle performlogical switching functions as fast as is physically possible, and thereis also no expensive optical-electric-optical (“OEO”) conversion processrequired to link them with present-day optical telecommunicationssystems. To date, however, they have not been feasible largely onaccount of lack of scalability and the absence of fully scalable AND-and NAND- and other logic gates. The invention disclosed here makes upthis lack.

SUMMARY OF THE INVENTION

It is an object of this invention to provide new digital 2-level andmulti-level simplification methods using vector logic.

It is another object of this invention to provide plans for all-opticalprocessors and circuits using vector logic.

It is yet another object of this invention to provide plans for opticalMultiplexers, Demultiplexers, flip-flops, AND gates and other similardevices

The invention is directed to apparatus, methods, systems and computerprogram products which permit a simplification of the logic required forperforming a certain function to a minimum set of logical elements ofoperations. This permits the complexity of digital circuitry to bereduced and the speed of the computation to be increasedcorrespondingly. It also improves reliability and diminishes materialcosts.

The logical application is accomplished using a system of propositionallogic in which propositions are represented as vectors or displacementsin a space. This is applied to the simplification problem, the problemof finding a method for reducing logical schema to a shortestequivalent. Specifically, an exemplary novel feature of the presentinvention includes placement of an origin within a Boolean cube, andthus making the Boolean cube into a vector space in which vectors can betranslated, added and multiplied, rather than a static structure of bitrepresentations in 1s and 0s.

The techniques given here are applied to the simplification orminimization problem, the problem of finding a method for reducinglogical schemata to their shortest equivalents, in both two-level andmulti-level forms. Applications of the system of vector logic toproblems of electronic circuit minimization, to free-space opticalprocessing, to flat-optical processing, to logical processing usingcolor images, and to the design of logic gates, including the AND-gateand NAND-gates, using a polarization implementation of the vector logicare described. The polarization implementation is fully scalable as itrequires only four passive elements: beamsplitters, reflectors,polarizers and retarders.

The present invention provides various exemplary methods forimplementing the vector principles for propositional logic for opticalcomputation. The first exemplary implementation is flat-optical, whereina Mach-Zehnder interferometer-like device is used to operate in eitherSOP or POS-form, which is the basic “cell” in the Karnaugh map sense foran all-optical processor. AND-, XOR-, XNOR- and NAND-gates may beconstructed from these cells using the vector optical implementation. Asecond exemplary implementation includes a system wherein the logic isimplemented in sequences of spatial light modifiers. In a thirdexemplary implementation of the present invention, basic principles ofcolorimetry are used for a simple colorimetric optical processingsystem. A fourth exemplary implementation of the present inventionincludes a Mach-Zehnder “cell” used for a device which does not dependon a bistable ‘and’ sum sigmoid filter.

In general, in one aspect, the invention features a novel logic system.

In another aspect, the invention features an optical implementation ofthe novel logic system to produce all types of optical logic circuits.

Additional advantages of the present invention will become apparent tothose skilled in the art from the following detailed description ofexemplary embodiments of the present invention. The invention itself,together with further objects and advantages, can be better understoodby reference to the following detailed description and the accompanyingdrawings.

BRIEF DESCRIPTION OF THE DRAWINGS

The objects, features and advantages of the system of the presentinvention will be apparent from the following description in which:

FIG. 1 represents two-dimensional space for propositions.

FIG. 2 illustrates the propositions p and p v q. in the space of FIG. 1.

FIG. 3 is a diagram of the vector two-dimensional space showing theconjunctional normal schemata normal schemata or CNS-plane.

FIG. 4 is a diagram of the vector two-dimensional space showing thealternational normal schemata or the ANS-plane.

FIG. 5 is a diagram showing modus ponens in the CNS-plane.

FIG. 6 is a diagram showing modus tollens in the CNS-plane

FIG. 7 is a diagram showing the disjunctive syllogism in the CNS-plane.

FIG. 8 is the diagram showing how the ANS- and CNS-planes relate.

FIG. 9 is a diagram illustrating a simplifying operation within theANS-space.

FIG. 10 shows an extension of the ANS-plane of FIG. 4 to athree-dimensional ANS-space.

FIG. 11 shows an extension of the CNS-space to three dimensions togetherwith a hypothetical syllogism.

FIG. 12A illustrates a hypothetical syllogism with three variables inthe CNS-space.

FIG. 12B shows a view of the hypothetical syllogism in thethree-dimensional CNS-space.

FIG. 13A illustrates a cancellation technique used in simplifyinglogical representations and in accordance with the invention.

FIG. 13B shows the representations of FIG. 13A in graphical form.

FIG. 13C illustrates implication and equivalence.

FIGS. 14A, 14B and 14C illustrate a solution to the simplificationproblem using the techniques of the invention.

FIG. 15 shows a 4-clause schema simplified.

FIG. 16A shows a 3-clause schema simplified.

FIG. 16B shows the truth-table for the representation of FIG. 16A.

FIG. 17 shows a 4-variable vector diagram simplification.

FIG. 18A is a diagram illustrating the Fix Rule for d=2.

FIG. 18B is an illustration of an example of the Fix Rule.

FIG. 19 is an illustration of the Fix Rule for d=3.

FIG. 20 illustrates application of the invention to situations in whichdeveloped normal formulas are not the point of departure.

FIG. 21 illustrates the equivalence of a developed alternational normalform and its undeveloped counterpart.

FIG. 22 illustrates the simplification of an undeveloped set ofstatements.

FIG. 23 illustrates another simplification of an undeveloped set ofstatement taken from Quine.

FIG. 24 illustrates an equivalence within the set of statements shown inFIG. 23.

FIG. 25 illustrates the Consensus Theorem.

FIG. 26 illustrates the dual of the Consensus Theorem.

FIG. 27 illustrates a superfluity shown in FIG. 23.

FIG. 28 illustrates a target circuit to be simplified in accordance withthe invention.

FIG. 29 shows a simplest circuit equivalent to the target circuit.

FIG. 30 is an illustration of optical computation of modus ponens.

FIG. 31 is an illustration of interferometric processing for modusponens.

FIG. 32 illustrates an optical element used for disconjunction andconjunction in a free-space optical processing.

FIG. 33 is an illustration of flat optical processing.

FIG. 34 is an exemplary optical AND-Gate in accordance with the presentinvention.

FIG. 35 illustrates the Sigmoid Characteristic for an AND-Filter inaccordance with the present invention.

FIG. 36 is an exemplary optical XOR-Gate in accordance with the presentinvention.

FIG. 37 is an exemplary optical XNOR-Gate in accordance with the presentinvention.

FIG. 38 is vector representation of an exemplary optical NAND-Gate inaccordance with the present invention.

FIG. 39 is an exemplary optical NAND-Gate in accordance with the presentinvention.

FIG. 40 illustrates an exemplary implementation of an exemplary CNS-cellin accordance with the present invention.

FIG. 41 is a chart of vector logical equivalents of “If p then q”.

FIG. 42 shows a computation of modus ponens using SLM implementation.

FIG. 43 is a view of an SLM device embodying the functionality shown inFIG. 42.

FIG. 44 is an illustration of the same vector addition utilizingsequences of SLMs.

FIG. 45 is an illustration of colorimetric computation of modus ponens.

FIG. 46 is a colorimetric simplification of pq v p q.

FIG. 47 shows the colorimetric vector space.

FIG. 48 is an illustration of the simple colorimetric sum device.

FIG. 49 is an elaboration of the device shown in FIG. 48.

FIG. 50 shows the wavelength bands of the filter for the device shown inFIG. 49.

FIG. 51 is a 3-D AND-cell using polarization and vector position foroptical computation.

FIG. 52 is the hookup of the device in FIG. 51 to the next input gate.

FIG. 53 is a COIN (coincidence) cell based on FIG. 51.

FIG. 54 is a half-added based on FIG. 51.

FIG. 55 is a MUX based on FIG. 51.

FIG. 56 is a (pNANDq)NAND(pNANDq) gate for p and q input based n FIG.51.

FIG. 57 is the same, for p and −q input.

FIG. 58 is an exemplary Polarization NOR-gate.

FIG. 59 is an exemplary NOR-gate for −p−q input.

FIG. 60 is the same for p−q input.

FIG. 61 is the same for −pq input.

FIG. 62 is the same for −p−q input.

FIG. 63 is the schematic for a wholly conservative logic polarizationcorridor for the AND-function.

FIG. 64 shows a Conjunctional Normal Form Truth-Table.

FIG. 65 is a MUX using the principle of FIG. 63.

FIG. 66 is a flip-flop for all possible paths using the principles ofFIG. 63.

FIG. 67 is the same, for S=1, R=0, present Q=1.

FIG. 68 is a spatial path-coding for p, q, −p, −q in the propagation ofrays through a short fiber.

FIG. 69 illustrates an optical AND gate and switching device usingpolarization photochromism.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

In the following description, for the purposes of explanation, numerousspecific details are set forth in order to provide a more thoroughunderstanding of the present invention. It will be apparent, however, toone skilled in the art that the present invention may be practicedwithout these specific details.

Part I describes a system of propositional logic in which propositionsare represented as vectors or displacements in a space. Part II givesthe application of the system to the simplification problem, the problemof finding a method for reducing a truth-functional schemata inalternational normal form to a shortest equivalent. Part III is aboutapplications: (i) to problems of electrical circuit minimization; (ii)to free-space optical processing; (iii) to “flat” optical processing;(iv) to logical processing using colorimetry; and (v) to polarizationbased processing.

Part I

Imagine a space in which the co-ordinates from the origin 0 arepropositional addresses or possibilities, for example as depicted inFIG. 1. Let (1, 0) be the propositional address p, and (0,1) thepropositional address q. Then (1, 1) is the point p, q, and we can letthe sign “+” be an operation on p and q which is defined by distance anddirection from the origin, by which ‘p+q+p+q’ is an instruction to gotwo units in a p-ward direction, and two units in a q-ward direction.The operation performed by someone obeying this instruction iscommutative and associative.

We can now represent the proposition that p as a directed line-segmentor vector along the p- or x-axis from the origin 0 to the point orpropositional address p in the space, representing the vector p inboldface as is standardly done to distinguish it from the possibility pwhich is represented as the point at the arrowhead of p. We can also letthe vector q be the proposition q, represented in the space as thevector pointing straight up the q- or y-axis to the point q.

Now if we build up the space interpreting the operation “+” as “v”, thex- and y-axes will obviously represent lines of logical equivalence. At(2, 0), or p, p, for example, we will find the arrowhead of p v p, andat (3, 0) the arrowhead of p v p v p. These and the rest of theproposition vectors along the p-axis are logically equivalent to thebase vector p. At (0, 2), or q, q, we will find the arrowhead of q v q.Thus p and q will stand in for the usual unit vectors i and j. (Thevector space of propositions can however have infinitely many directionssuch as s, t, u, v . . . , which will become important later on when atechnique is given to simplify propositions with large numbers ofliterals.)

We are also now in a position to represent the proposition p v q in thespace as p+q, the vector resultant of the vectors p and q, which travelsfrom the origin to p, q. Then p v q is itself a vector.

I will call a vector diagram for the propositional calculus such as FIG.2 a V-Diagram for the proposition or schema. The V-diagram can befurther built up by adding the negation symbols for the negative vectorsp and q in the negative or reverse directions along their respectiveaxes. So we arrive at all of the literals, which are single letters andnegations of single letters, and we can also find pairs of singlenegated or unnegated letters, the propositions p v q, and p v q.

Let us call the vector two-space in FIG. 3 the CNS-plane for the planeof the “conjunctional normal schemata”.

We are now free to explore another plane, for example as illustrated inFIG. 4, the plane of the alternational normal schemata, or theANS-plane, in which the points are not alternations but conjunctions,and the vector operation “+” within the space is interpreted asalternation. The ANS-plane and the CNS-plane are duals, so that eachpoint in each plane correspond to its dual in the other plane. This alsomeans that the uniting operation in the CNS-plane is related to the dualof the operation in the ANS-plane, and vice versa. In the CNS-plane “+”is alternation, and so in the ANS-plane it is conjunction. The operation“→” in “α→β” in the CNS-plane is to be read as implication or theassertion of the conditional. In the ANS-plane “→” is to be read as thedenial of the negation of implication, which is the denial of theconjunction of the antecedent with the negation of the consequent. Thewhole ANS-plane is to be read as a systematic set of denials, thedenials that the propositions given at the base of the vector arrowheadsimply a contradiction. This will be obvious if we remember that arrowsending at the origin rather than those issuing from it, as in theCNS-plane, are assertions in the ANS-plane.

In both of the planes certain familiar truths appear as expressions ofthe main principle which governs “+” or vector addition, the so-calledparallelogram law of Galileo. In the CNS-plane we can think of thepremises of an argument as component vectors, and the resultant as theconclusion. Then an elementary valid argument-form in the CNS-plane is aparallelogram starting at the origin 0 in that plane. The conjunction ofthe alternations yields the conclusion, and we get modus ponensappearing as in Diagram 5. If the vectors are represented asdisplacements around 0 in the V-diagram, the modus ponens in theCNS-plane is the set of displacements

p q Premise 1 −1 1 + p v q Premise 2 1 0 p Conclusion = 0 1 q

Modus tollens, as illustrated in Diagram 6, appears as:

p q Premise 1 −1 1 + p v q Premise 2 0 −1 q Conclusion = −1 0 p

The disjunctive syllogism, as illustrated in Diagram 7, appears, withits displacement matrix, as:

p q Premise 1 1 1 + p v q Premise 2 −1 0 p Conclusion = 0 1 q

Consider now the relation between the CNS-plane and the ANS-plane. Thereclearly is one, as they share the literals and the all-important origin0. The two planes can be brought into harmony if we represent them,arbitrarily, as lying above and below the origin in a space whose thirddimension runs along the conjunction-alternation axis, puttingalternation at the top and conjunction at the bottom.

The result is a space, or the part of it near 0, with two planes aboveand below the origin. The origin 0 appears in the vertical axis betweenthe two planes. The whole space of FIG. 8 generates further principlesof the propositional calculus. Take p v q in the top right hand corner.Negating it comprehensively, in all three dimensions, or developing itthrough the origin, gives the point p q in the ANS-plane. This is one ofthe two forms of DeMorgan's theorem. Its other form can be found bycomprehensively negating pq in the ANS-plane, and travelling through 0to p v q in the CNS-plane. The CNS-/ANS-space as a whole has anintriguing and beautiful structure, as it combines the dimensions ofalternation and conjunction, the various propositions formed from atomicp and q, and the dimension of negation.

Operations within the ANS-space have “+” representing conjunction. Whenall the non-equivalent conjunction points are established in the space,pairs and other combinations of the given points or conjunctions aregiven as alternations or vectors. So we get a resultant of pp from pq vp q by relating the two vectors to the origin 0 in a parallelogram (FIG.9).

In the CNS-space, on the other hand, the corresponding operationproduces alternations, and the operation within the space which combinesthem is conjunction. So we get sets of conjunctions, e.g. thoseimportant ones involving p v q, which are among the more importantarguments of natural deduction.

Assume now in the ANS-space a third proposition r, and a third dimensionz in which the unit vector r is to be found. So we get the r-plane, theone swept out by the vector r. This space can also be represented in twodimensions on the page. In FIG. 10 the negation-affirmation axis, whichfollows the z-axis in the order of rotation of the variables about 0, isinserted to prevent the occlusion of lines and points.

To check the validity of the hypothetical syllogism (−1, 1, 0) (0, −1,1), (−1, 0, 1), with three variables, in the CNS-space, we can representit as in FIG. 11.

As illustrated below, and in FIGS. 12A and 12B, the of the validityargument appears, using three sets of coordinates, as

p q r Premise 1 −1 1 0 + p v q Premise 2 0 −1 1 q v r Conclusion = −1 01 p v r

Note the simplicity of the given representation or perspective on thehypothetical syllogism in FIG. 11, matched only by the simplicity of thepqr string −1, 1, 0, 0, 1−, 1, −1, 0 1, which is merely a set ofinstructions for displacements in a 3-space.

The vector system can be used in the CNS-space to display otherprinciples, for example implications, by which p v q implies p→q. Italso shows that p v q implies 0→ p v q, as well as q→ p and p v q→0.Furthermore, the vector system shows nicely the principle of materialequivalence, which states that (p→q)(q→p) is equivalent to p

q (FIG. 13).

The starting point of all these vectors, together with the direction,gives the end point. “Together with” here means treating the points anddirections algebraically as themselves directions from the origin. Thisyields a cancellation technique in which a starting-point of 0 iscancellation of no literal, and an end point of 0 is the cancellation ofall the literals

Starting End Point Direction Point 0 p v q p v q p p v q q q p v q p p vq p v q 0

Parallel principles can be given for the ANS-plane. Here “α→β” means thesame as in the CNS-plane; which is α⊃β, the conditional, but the reasonis hard to see, though interesting. Take the proposition p→q in theCNS-plane. It is represented by (among others) an arrow from the point pto the point q. In the ANS-plane we find an arrow from p to q. Call itv. But what does v mean? Note that the CNS-plane vector from p to q istrue if v is false. For v is p q, and the same vector or direction as pq→0. If we want the ANS-plane vectors to represent truth, we must readthem as the denials of the conjunction of the proposition p at the baseof the arrow with the negation of the proposition q at the arrowhead, or−(p q). Each arrow in the ANS-plane then reliably represents aconditional.

This reveals something further about the all-important 0, the origin. Wehave just learned that in the ANS-plane an arrow from p to q is p q, tobe read however as a negation. So what does α→O mean? O is p p. So theconjunction of α and −(O) of a α−(pp). But this is α( p v p), which isequivalent to α.

Similarly, in the CNS-plane, all the arrows which depart from 0represent an instance of α v β, where α is 0. Take an arrow from 0 to pv q. 0 is the tautology p v p. The negation of this p p, and so thearrow to the point p v q is −(p v p) v pq. But the first disjunct ofthis is equivalent to p p, and so it is always false. Hence thealternation is equivalent to the second disjunct, or the assertion pq.

If a vector in the ANS-space is directed towards 0, 0 has the effect ofreversing the truth-values of the base propositions. Moving towards 0from the base (p,q) in the CNS-space we get the vector p v q. 0 has theeffect of putting p and q through the Sheffer-function “|”. The vectormoving away from 0 in the CNS-space towards e.g. (p,q) is also thevector from the base (p,q) to 0, and so it is the vector p v q or p v q.

In a dual fashion, if we are moving towards 0 in the ANS-space, we getthe base values, so that pq→0 is pq. From 0, a vector to ( p, q) willthus be pq. In the ANS-space 0 has the effect of putting p and q throughthe dagger function “⇓”, by which p⇓q is p q.¹ ¹Wittgenstein's operatorN in the Tractatus could be described as a generalization of ⇓ to morethan two places, as N(p,q,r), for example, is pqr. We could alsodescribe a generalized Sheffer operation for more than two places whichtrANS-forms a base such as say (p,q,r,s) into p v q v r v s. Thisoperation could be called S for “Sheffer”.

Part II

The simplification problem is the problem of reducing truth-functionalschemata (or, in the system I am describing, systems of vectors in theANS-space) to their shortest equivalents. A practical method for doingthis, in alternational normal form continues, as Quine observes (Quine,1982, p. 78), to be suprisingly elusive.

In the ANS-space “vector logic” can be applied to the problem in thefollowing way. Take the schema pq v p q, which as well as implying p isequivalent to p. To simplify it, form the parallelogram from the origin0, pq and p q to the resultant or vector sum point. Call it

, for “implicant”. The vector acting at

, which is in this case pp, implies pq v p q. So

splits up alternationally, into its components, pq v p q, towards theorigin.

Next note that pq is equivalent to pqp, so that the arrowhead at pq canbe dragged to pqp. But pp can also be dragged to p. Now we have an arrowfrom p to pqp. But this arrow an be translated into a position on top ofthe arrow from p q to pp. The same procedure yields a double-headedarrow between pq and pp, and the result can be read as pq v p q

p.

When an implicant splits up into its alternations towards the origin, ifthere is a proposition σ (for “simplest equivalent”) at the center ofthe parallelogram formed by 0, the disjuncts of a two-clause targetschema, and

, then σ is a shortest equivalent of the target schema. But this onlyworks for pairs of schemata which do have an

-point.

The general simplification procedure, in the ANS-space, is as follows.

(1) Represent the alternational normal schema, the target schema t, as aset of vectors in the ANS-space. Each clause or disjunct of t is aposition vector (i.e. one pointing to 0) with 0 at one corner of aparallelogram made of propositional addresses to the

-point at the other. Any two other outside vertices of such aparallelogram are implicants

which are among the original clauses of t.

(2) Pick any two clauses. If there is a propositional address σ at themidpoint between the component clauses, the vector from

to σ, i.e. σ, is the simplification of and can replace the relevantclauses of t, as in the case where t is pq v pq,

is pp and σ is p.

(3) Generate

-implicants until each clause or vector has been used at least once. Ifa disjunct d of t cannot be used because it forms no propositionaladdress with any other disjunct, then d must appear unmodified in thefinal schema which is the simplification of t.

(4) If an

-point exists in t, delete the vectors which produce it in favor of thevector from

to 0.

(5) For a clause in a schema which subsumes another clause, e.g. pqr vpq, eliminate the subsuming clause, in this case pqr, leaving pq.Implications arising from subsumption can be written into the wholevector system of t as components where relevant. For example, an arrowcan be drawn from pqr to pq in the above example.

Rule (5) applies for example to pq v p, which is an undeveloped orunbalanced schema in which pq subsumes p. How does pq v p simplify to p,when it seems to yield p v q? The Answer, which cashes the metaphor of“subsumption”, is that p really represents a plane, in a 3-space,sweeping out the whole p-domain, or any ANS-schema with p in it. So itis a kind of type fallacy to represent pq alongside p in a single schemaas if they were to be treated separately. For pq, and p q, are really“elements” of p itself. A cube is not so many faces and so many lines,but it can be represented as lines producing faces or vice versa. As amatter of philosophy, therefore, vector logic can avail itself, as rule(5) does, of a preliminary use on Quine's operation (i) from “A Way toSimplify Truth Functions”, which has us ‘drop the subsuming clause . . .if one of the clauses of alternation subsumes another . . . ’. Quine'soperation (i) also replaces α v αφ with α v φ, and the same for thecorresponding α-schemata (Quine, 1955, p. 627).

(6) Couples such as pq v pq or pqs v pq s cannot be summed to zero, theorigin

(7) Translate vectors as in FIG. 14. Any superpositions of parallelarrows in opposite directions represent equivalences. (a) Drop thelonger clause at the end of any double-headed arrow. (b) Drop pairs,triples etc. of double-headed arrows which meet at a point in favor ofthe vector from that point to 0. (c) Drop a vector or clause in thetarget schema which is itself the resultant of any other two vectors.

(8) A simplification is complete if in the system which replaces thetarget schema: (a) no vectors or clauses are subsumed by others (seeRule 5); (b) no double-headed vectors remain, or, in other words, if allequivalences in the system have been exploited.

Take next the simplification of the four-clause target schema pqr v pq rv pqr v p q r. The first job is to plot the target schema in aV-diagram. We get two parallelograms, with two i-points, qrqr and p rpr, and two σ-points, qr and p r, which are final in the sense that theydo not generate a further σ-point. Hence the target schema is equivalentto qr v p r.

Now take the simple-looking three-clause schema pqr v pq r v p qr. Theresultant is pq v p r.

Here the σ- and

-points function as before. But something else has happened. The vectorpq r has been used twice, once along with pqr to give pq, and again,with p qr, to give p r. Why was pq r not exhausted by its first use, andwhy can it be used again? The Answer can be seen by looking at thetruth-table for pqr v pq r v p qr, which is

1. pqr T 2. pq r T 3. p qr 4. p q r T 5. pqr 6. pq r 7. p qr 8. p qr

Truth, it could be said, is not exhausted by use. The p r of line 2 isso to speak redundant, as line 2 has already been captured by thedisjunct pq, and so line 4 has had half of its work already done.

This simplification procedure is theoretically an improvement on thetechniques used in Karnaugh maps (Garrod and Borns, 1991, p. 153 ff.),as it needs no wrapping around and can be used mechanically and easilyon more than four variables—any number fits into the “propositioncircuit”, which gradually turns from a square, with two variables, intoa hexagon, with three, and finally into a circle, with an infinitenumber of variables. With four variables, the logical space is as givenin FIG. 17.

The whole figure in FIG. 17 is a “measure polytope” or hypercube, thoughone with a further complex internal structure. There is no limitation oftessellation to the number of propositional variables or vectors p, q, rs . . . that can be handled, because the space is derived not from aclosed figure, such as a cube, but from a sheaf of lines in thegeometrical sense. Not all closed figures tessellate. All the lines ofthe multi-dimensional sheaves are coincident.

Consider in FIG. 17 a simplification from pqr s v pq rs v pq rs v pqr sv p qrs v pq rs to p rs v q s. This proceeds as shown, with the sixvectors reducing to two. The first vector or Φ-point in thesimplification, p rs, results from the implicand pair pq rs v p qrs,dropping the up-down q/ q component. In this case we have asimplification from two four-letter schemata to one three-letter schema.The remaining four schemata are all needed to fix the Φ-point q s. Bothpairs pqr s v p qr s and pq rs v p qr s must give a fix on the sameΦ-point if the reduction from four literals to two is to be justified.For either pair by themselves is not sufficient for the requiredbiconditional. The general rule isV=2dwhere v is the number of vectors required to make the fix on theΦ-point, and d is the drop in the number of literals from the clauses ofthe given schema to the resulting clause in the target schema.

Another illustration of the Fix Rule is pqr v pq r v p qr v p qr v pqr vpq r v pqr, which is, however, equivalent merely to p v q v r, as itcovers every line of the truth-table except pqr.

One pair of implicands is p qr v pq r, which give the Φ-point p. But asthis is a drop down from three letters to one, we need a fix of fourvectors or two vector sums on the point, and the third and fourthvectors pqr and pq r provide it. The same sort of fix appears with q(pqrv pq r and pqr v pq r) and r( pqr v pqr and pqr v p qr).

A much simpler though negative example of the Fix Rule is pq r v p qr,which seems to give p as an Φ-point resultant, but fails to for lack ofa fix on the point p, as four, not two vectors must converge on it forthe drop. This acts as a constraint on the vector arithmetic. We seem toget

p q r 1 1 1 + 1 −1 −1 = 1 0 0

But the Fix Rule rules this out. If x columns are filled with numbers,positive or negative, then the number of non-zero columns in the summust be x−1. The Fix Rule will seem entirely unartificial when onerecognizes that what it means in, say, a 3-space, is that a literal orone-letter proposition is a face, and so four corners are needed todetermine it. A two-letter proposition is a line, and so only twoletters are needed to fix it. And a point in a 3-space is a three-letterproposition.

Consider as another illustration of the Fix Rule pqrs v pqr s v pq rs vpq rs v p qrs v p qr s v p qrs v p qrs. This is very obviouslyequivalent to p, and since d=3 for each clause, v for the point p=2d. Sod=8, and eight vectors or four vector sums are needed for the fix on theΦ-point.

In many cases the target schema is unbalanced in the sense that itsclauses have different numbers of conjuncts and so they need to be putinto developed alternational normal form. An example pq v p qr v pqr(Quine, 1982, p. 75). This is equivalent to pq v pr v pqr. Like theearly Quine's procedure in “The Problem of Simplifying Truth Functions”(Quine, 1952, p. 524), the vector simplification method given so far hastaken the cumbersome ‘developed normal formulas as the point ofdeparture.’

If t is developed uniformly we get pqr v p qr v pqr v pq r r, which in aV-diagram is clearly pqr v pq v pr, as pr lies midway between pqr and 0,and pq lies midway between pqr and pq r. But without development, we cantake the ∴-point for p qr v pq, which is pr, and argue that since pr→pqr v pq (where “.expression” and “expression.” represents bracketing ofthe “expression” that precedes or follows the dots), and pr subsumes pqr, the longer p qr can simply be replaced by its own implicant.

The equivalence of undevelopedpq v p q v qr v q r  (i)andp q v pr v q r  (ii)is harder to establish. It is one which resists as many as twelve fellswoops (Quine, 1982, p. 76, also in 1952, pp. 523-527) or shortertruth-tables. In the vector space with developed alternational forms theequivalence is easy enough to see. The developed form of thisequivalence is easy enough to see. The developed form of this exampleis: pqr v pqr v pq r v pq r v p q r v p qr.

The same result can be obtained using column matrices for the pairs ofvectors. Then for pqr v pqr we get

p q r −1 1 1 + pqr −1 −1 1 p qr −1 0 1 prAnd for pq r v pq r we get

p q r −1 1 −1 + pq r 1 1 −1 pq r = 0 1 −1 q rSimilarly, for p q r v p qr we get

p q r 1 −1 1 + p qr 1 −1 −1 p q r = 1 −1 0 p q

It should be noted that in this example too the Fix Rule applies. Itwould be nice to take the vectors in a different order, so that p qr andpq r are chosen instead of p qr and p qr, and also pq r and p qr insteadof pq r and pq r. This would yield r and p instead of pq and q r in thewhole system. But that would mean dropping from three letters to one inthe case of these two pairs of alternations, and we cannot do that asthere is no fix on r or on p.

There may of course be more than one “shortest” schema. In Quine'sexample there is obviously is on inspection a second. The vector systemqr v pq v p r has the same overall “effect” in the vector-logical space.

Let us now try this example with the use of the

-points, the key prime implicants. Take first pq v qr. This alternationis implied by the

-vector which forms the parallelogram with 0. But there is no Φ-point,and so, apparently, pq v qr is not equivalent to pr. Yet in the contextof the whole scheme p q v qr v pq v q r, it is. To see this, we move thefree vectors p q and q r from the right-hand side of the V-diagram tothe parallelogram on the left. The implicand of p r, which is p q v q r,slides into place from p q v q r back to pr, and the two-way implicationor equivalence is established.

The vector summation of p q and qr to p qqr or p qr is disallowed by theFix Rule, according to which the number of vectors needed to make a fixis equal to the d-th power of 2. This summation would actually produce anegative value for d. As the number of literals rises from two to three,the drop increases from 2 to 3, or −1.

Quine gives another interesting example of a simplification with foursimplest equivalents, one which also illustrates the method ofsimplification for non-developed or unbalanced schemata like the lastexample. The example (Quine, 1952, p. 528) is pqr v p r v pq s v pr vpqrs (FIG. 23).

We begin by generating vector sums for the various disjuncts. We can seefairly easily that pr and pqrs to start with, yield a parallelogram, butit seems to end at an

-point outside the logical space. Yet if we study that point, we can seethat it is actually at the co-ordinates pqrspr. This point, however,contains a contradictory or backward and forward instruction, namely ther from pr and the r from pqrs which can both be deleted. There is also adouble p in the final address, and one p of these, but not both, can ofcourse be deleted. This leaves an end-point for a vector pqs. By similarreasoning, we can arrive at the vector qrs as the vector sum of p r andpqrs. And similarly pqr with p r gives pq, or pqr with pr gives the

-point qr. Each vector must be used at least once if it is not to appearunchanged in the simplified schema.

The corresponding Φ-points, however, do not appear at the designatedaddresses which are shortened versions of their

-points, and so the equivalence of the

-points and their implicands is not established. Just as in the exampleshown in FIG. 20, the drag-back effects described in connection with pand pp in FIG. 14 do not apply.

As before, in FIG. 24, we first write in the parallelogram from 0 forthe pair pqr v p r. This gives an

-point at pqp, and so from pqp we write in a pair of vectors to pqr andp r. Now pqr implies pq, and is subsumed by it, and we can represent thesubsumption rule here by drawing in the vector from pqr to pqp. FIG. 24is now showing an equivalence between pqr and pq, but only in thepresence of the p r in the alternational schema pqr v p r, i.e. with thetranslated or “borrowed” vector p r→0.

It is worth realizing that in the reductions in developed normal form,the implicands can be replaced by the implicant only because of thevarious biconditionals or double arrows at work. There is no intrinsicmagic in the Φ-point. In the undeveloped examples, too, clauses do notdisappear in a general way because pairs of disjuncts collapse intotheir implicants, but because of the presence of specific conditionselsewhere in the schema, which translated have the effect or creatingbiconditionals.

Finally, why is pq s superfluous in the example given in FIG. 23? TheAnswer is interesting and complicated, and principles about superfluityneed to be established.

Take the truth, sometimes known as the Consensus Theorem, that pq v p rv qr.

.pq v pr. Representing this in the ANS-space for p, q and r, we can seethat the implicant qr is the resultant of the disjunction of pq and pr(FIG. 25). We can give it as a general truth that implicants, in theANS-space, are resultants.

Let the left-hand side of the Consensus Theorem be represented asqp v pr v qr

The Theorem says that the disjunct qr is superfluous. Consider the dualof the left-hand side of the Theorem, in the CNS-space. It is(q v p)( p v r)(q v r)

This is the conjunction ( q p)(p r)( q r). But clearly the last conjunctis superfluous, as the first two conjuncts imply it by a hypotheticalsyllogism, in the sense that if they are true, so is it (FIG. 25).

It is nice to see the dual roles of conjunction and alternation, orANS-and CNS-spaces, truth and falsehood, and how the concept of theresultant and the component binds them together.

In the following truth-table, we can that the (q r) resultant is so tospeak “covered” by its component with respect to truth, in theANS-space, and falsity in the CNS-space. That is, with the disjunctionsin the ANS-space the addition of an extra truth on already true lines ofthe truth-table does not affect the truth of the whole schema. Andsimilarly in the CNS-space, if the whole schema is already false, addinga false conjunct will not affect that result.

ANS- CNS- p q r qr v pq v pr (q v p) (p v r) (q v r) T T T T T F T T F TT F T T F F F F F T T T T F T F F F T T F F F F F F

We are now in a position to deal with the superfluity of pqs in Quine'sexample in FIG. 24. Disjunctive clauses in the ANS-space like pq s aresuperfluous when they are components. Before the representation of theschema pqr v pqrs v p r v pr v pq s with a view to simplification, wecan simply run a check to see if any of the clauses are implicants oriota-points for any others. We can easily find that pq s→.pqr v p r fromthe truth-table for the schema; all the lines on which pq s is true arealso lines on which either pqr or p r is already true, and so pq s canbe deleted from the schema to be simplified.

Geometrically, the construction is as follows: (FIG. 27). Note that pq sextends to pq sp. This is however the implicant for pqr s v p r.However, pqr s itself extends to pqr spq. This last schema is theimplicant for pqr v pq s. So pq s gives way to p r v pqr s. But pqr scan itself can be dropped in favor of pqr v pqr s. Any line of thetruth-table for pqr on which is true is also one which either pqr or p ris already true. Hence pq s can be dropped.

So for Quine's example in FIG. 23 we are left with the fourpossibilities:

These examples, and others like them, suggest the possibility of furtherapplications of simplifying geometrical theorems and methods to thesimplification problem.

The charm of a vector simplification technique is that is follows aleast-action principle, for any number of propositional vectors, in thesense that the problem is not one of finding shortest equivalents totruth-functional schemata. Rather the space, inasmuch as it is fixedvector space in which all free vectors having the same direction are ina sense the same directional vector, is unable not to give the desiredresult.

As to propositional logic as whole, it is nice to have all of thenineteen or however many clanking “rules of inference” within the space,so that there is just the one intuitively obvious method of argument:vector addition. It is really absurd to think of empty or “formal” rulessuch as association and communication as having the same status as saymodus tollens, which is genuine “motor” that advances arguments throughlogical space. Association and commutation should flow out of the natureof the logical space, and in the vector space they do. The vectors p v qand q v p, for example, have the same end-point, though they arrive atit by different but corresponding routes.

Part III

(i) Electrical and Integrated Circuit Minimization

Let us now see how the techniques described can be used in a routine forsimplifying electrical and integrated circuits. Take the target circuitABC+A C+AB D+ĀC+ ABCD (FIG. 28).

The first job is to plot this in the ANS-space as the set of vectors pqrv p r v pq s v pr v pqrs, as in FIG. 23 above. Following theabove-discussed general simplification procedure, in the ANS-space, wecan simplify this system of vectors to e.g., pq v pr v pr v pqs. Theresultant schema can then be translated into the circuit diagram AB+AC+ĀC+ ABD, as illustrated in FIG. 29.

Note that the target circuit, as illustrated in FIG. 28, has five gates(G=5), that the total number of inputs into these gates is twelve(I=12), and that the redundancy factor (i.e., the number of times anoriginal input is used again, corresponding to the join dots) is seven(R=7). These figures drop to G=4, I=9 and, most importantly, R=2 for thesimpler circuit, as illustrated in FIG. 29, representing correspondinggains in materials savings, speed and reliability.

(ii) Free Space Optical Computation

More than ten years ago the National Academy of Sciences Panel onPhotonics Science and Technology Assessment declared that ‘The ultimatebenefit of photonic processing could occur if practical optical logiccould be developed’ (Whinnery et. al., Photonics, 1988, p. 35). So farthe implied challenge of the Panel has not been met.

Vector manipulation has been one of the big success stories for opticalcomputation, but vector techniques themselves promise an application tothe logic of optical computation as a whole. The full ANS-/CNS-spacecould be built as an optical device for checking the validity ofarguments or as a logic device for optical computation, and also assimplification machine. Each operation in the space is a laser, and theresultant proposition-points such as p and pq and pqr are multifacetedbeamsplitters or mirrors which reflect the beams in the correct logicaldirections at the correct logical strengths to ensure the requiredimplications.

Thus in FIG. 30 a beam V can be sent from the origin to a half-darkeningbeamsplitting mirror at the node p. At p it is split and sent athalf-strength to q, and to q. Simultaneously, a second beam U from 0 issent to the node p v q, which is also p→q. At this point U is split andsent at half-strength to p and to q. The proposition p is said to“half-imply” q, in the sense that with one other proposition it doesimply q, and the proposition p v q is said to “half-imply” q in thesense that with one other proposition (p) it does imply q.

Both half-implication beams are coincident on q, and at q thephotoreceptor gives a reading of 0.5+0.5 or 1. The system has opticallycomputed modus ponens; from an input of p and an input of p v q, it hasyielded up q. The system gives a physical interpretation ofbeamsplitting as multiple implication and of darkening as fractionalimplication.

The same principles will apply to the other rules of inference andlogical equivalences.

A development of the system given for modus ponens in FIG. 30 obviatesthe need for a free beam for e.g., p to q, and simplifies the design ofthe node. In FIG. 31, the beam to p is split, at full-strength, to p'simplicants, which are p v q and p v q (ignoring tautologies). Thebeamsplitter at p v q itself directs the beam to q at onlyhalf-strength, and the desired computation is achieved.

We can also arrange that in an embodiment of the uninterpreted (p, q, .. . n) space, in which the base (p,q) is either p v q or pq (though notboth), configurations of the beamsplitters will allow the node to switchbetween the two states. A conjunctional state will correspond to aconcave configuration, as exemplified in FIG. 32A. In FIG. 32A, bothinputs are required for the activation of the node. An alternationalstate will correspond to a convex configuration, as shown in FIG. 32B.

(iii) “Flat” Optical Processing

An exemplary implementation of flat optical processing is illustrated inFIG. 34. Each position vector of cell 3400 is a laser source, p, q, etc.An input beam 3404 is sent from p to a combiner 3410 at O, and the samefor a second beam 3402 from the q direction. Intensity filters 3406 and3408 cut down the light from each source to half. The angle of thecombiner 3410 is set so that both of the now half-strength beams arecoincident on the output in the (arbitrarily chosen)−p direction. At theoutput there is an optical filter 3412 with a sigmoid characteristic sothat if the intensity of the beams is one or greater (no greater thantwo), then the output is one, and if the input is less than one then theoutput is zero, as illustrated in FIG. 35. Finally, at the output pointthere is a photoreceptor 3414, which announces that the twohalf-strength beams have converged on q for a value of 1. This systemwill be optically on iff p and q are input. Since this implementation,acts as pq or an AND-gate, the output beam can be used as a new p orinput for subsequent computations.

So far what has been described is a single gate. Gates of this type canbe combined, however, using the sigmoid-characteristic filter to controlthe output. Take next an XOR-function representing the logical schemap−q v −pq. In this case we can combine a p−q-gate with a −pq-gate, as inFIG. 36. Cell 3602 includes two input beams 3606 and 3610 at p and −qrespectively. Both input beams are passed through respective intensityfilters 3614 and 3616 in order to attenuate the signal in half. Bothinput beams are then combined at combiner 3620 and directed to sigmoidfilter 3628. Cell 3604 includes two input beams 3608 and 3612 at −p andq respectively. Both input beams are passed through respective intensityfilters 3618 and 3644 in order to attenuate the signal in half Bothinput beams q and −p are then directed to sigmoid filter 3626 by mirrors3622 and 3624 respectively. The outputs from each sigmoid filter 3628and 3626 pass through respective intensity filters 3642 and 3640respectively in order to attenuate the signals in half. The attenuatedbeams are then directed to sigmoid filter 3636 via mirrors 3630, 3632,and 3634. The output of sigmoid filter 3636 is detected by photodetector3638. These gates can of course be further combined for example to yieldan optical analogs of other integrated circuits, e.g. IC 74266, which isentirely composed of XOR-gates.

The XNOR-gate can be similarly represented, for example as a combinationof a pq-gate and a −p−q-gate as illustrating in FIG. 37. Cell 3702includes two input beams 3706 and 3708 at q and p respectively. Bothinput beams are passed through respective intensity filters 3714 and3716 in order to attenuate the signal in half. Both input beams are thencombined at combiner 3722 and directed to sigmoid filter 3628. Cell 3704includes two input beams 3710 and 3712 at −q and −p respectively. Bothinput beams are passed through respective intensity filters 3718 and3720 in order to attenuate the signal in half. Both input beams −q and−p are then directed to sigmoid filter 3626 by mirrors 3726 and 3724respectively. The outputs from each sigmoid filter 3628 and 3626 passthrough respective intensity filters 3642 and 3640 respectively in orderto attenuate the signals in half. The attenuated beams are then directedto sigmoid filter 3636 via mirrors 3630, 3632, and 3634. The output ofsigmoid filter 3636 is detected by photodetector 3638.

The NAND- or −(pq)-gate is more complicated in an interesting way. Itconsists of the three cells that represent those three lines of thetruth-table which negate the pq line. In the vector representation −(pq)or (by De Morgan's Theorem −p v−q) is given as illustrated in FIG. 38.

The optical implementation of −p v−q is illustrated in FIG. 39, theNAND-gate, which consists of three cells plus filters. Cell 3902includes two input beams 3908 and 3910 at p and −q respectively. Bothinput beams are passed through respective intensity filters 3916 and3918 in order to attenuate the signal in half. Both input beams are thencombined at combiner 3928 and directed to sigmoid filter 3938. Cell 3904includes two input beams 3912 and 3913 at −p and q respectively. Bothinput beams are passed through respective intensity filters 3922 and3920 in order to attenuate the signal in half. Both input beams q and −pare then directed to sigmoid filter 3940 by mirrors 3930 and 3932respectively. Cell 3906 includes two input beams 3914 and 3916 at −p and−q respectively. Both input beams are passed through respectiveintensity filters 3926 and 3924 in order to attenuate the signal inhalf. Both input beams −q and −p are then directed to sigmoid filter3942 by mirrors 3936 and 3934 respectively. The outputs from eachsigmoid filter 3938, 3940, and 3942 pass through respective intensityfilters 3954, 3952, and 3640 respectively in order to attenuate thesignals by ⅓. The attenuated beams are then directed to sigmoid filter3948 via mirrors 3950, 3944, 3946 and 3952. The output of sigmoid filter3948 is detected by photodetector 3638.

It is also easy to use the same techniques in the SOP—or ConjunctionalNormal Form. Modus Ponens states that if ‘If p then q’ and ‘p’ then ‘q’.As illustrated in FIG. 40, a CNS-cell will produce the output q if theinputs are p v −q and p. Cell 4002 includes two input beams 4004 and4006 at p and p v −q respectively. Both input beams are passed throughrespective intensity filters 4018 and 4020 in order to attenuate thesignal in half. Both input p is then directed to sigmoid filter 4014 bymirror 4008, whereas input p v −q is directed to sigmoid filter 4014 bymirror 4010 and combiner 4012. The output of sigmoid filter 4014 isdetected by photodetector 4016. As illustrated in FIG. 41, for p→q hasas translated or logically equivalent forms O→−p v q and p v −q→O.

A second method of exploiting the vector system for computation is moremarkedly spatial. Represent the propositions in the uninterpreted (p,q)space with spatial light modifiers (SLMs). When the first premise isinput, e.g. p v q, then the origin 0, and with it the position of thewhole space, are moved to the point p v q, or in a p v q direction. Wecould say the 0 becomes p v q, so that we are now in a p v qenvironment, a p v q world. Then p in the second SLM (FIG. 33) will beq, and we have modus ponens. And when the whole space is displaced in ap v q direction, q is p!

A second method of exploiting the vector logical characteristics oflight is more markedly spatial. The premises or inputs can berepresented in the CNS-space as spatial light modifiers. For example,let there be two modifiers 4202 and 4204, which represent the wholeCNS-space for two variables p and q as illustrated in FIG. 42A. Theorigin of the first modifier 4202, representing −p v q, is aligned withthe second modifier 4204, so that the origin 4202 is at −p v q on 4204,as illustrated in FIG. 42B. This represents a −p v q shift within 4204.We can say that if O becomes −p v q, then p becomes q, or even that whenwe are in a −p v q environment p is q. FIG. 43 shows an oblique view ofthe two SLMs in sequence in combination with a light source 4302,whereas FIG. 44 shows a side view. As illustrated in FIG. 43, lightsource 4302 incident on p on first modifier 4202 yielding q on thesecond modifier 4204. It is as if we are asking: if O is −p v q, what isp? The answer computed is: q. The results of the vector logic system forpropositional logic show that this implementation technique can be usedfor all the rules of argument, and for any number of variables.

(iv) Colorimetric Processing

Colored laser beams can be used so that the refractive angle is builtinto the vector rather than into the propositional nodes as the CIE(Commission Internationale de l'Eclairage) x-y chromaticity diagram (acolor mixing diagram) is itself a vector space. (Further a mixed systemof colored laser and colored mirrors could be used. Optical computationfor simplification may then merely include the colorimetric process ofadditive color mixing. For example, in the CNS-space, let p be red (R),p a complementary cyan blue-green (C), q a yellow (Y), q a complementaryblue (B), p v q yellow-red (YR) and p v q blue-red (BR). Also p v q isthe complementary of YR, a cyan blue.

The contradiction of 0 (the so-called “Nullpunkt”, or “white”)corresponds to the addition of complementary hues. For example, YR+BR=R,since Y and B are complementary.

With these colorimetric assignments we can compute modus ponens and theother rules of argument and truth-preserving substitutions. For example,as illustrated in FIG. 45, if proposition p is R from a light source4504 that passes through a red filter 4508, and if proposition p v q isCY from a light source 4502 that passes through a CY filter 4506, and ifq is Y, from colorimiter 4510, then p→q, or p v q, is CY. Together withR this give Y or q, as C and R are complementaries.

In the ANS-space we can perform simplifications colorimetrically. Forexample, in FIG. 46, exemplifies a basic simplification in which pq v pq is equivalent to p. Let YR, from yellow-red filter 4606, represent pq,and BR, from blue-red filter 4608, represent p q. The Y and B portionsof the beams cancel at colorimiter 4610 because they are compliments,leaving RR or R, which is pp or p.

The CIE xY chromaticity chart, for example as illustrated in FIG. 47, isa vector space. If colored beams are used for V, U, W . . . N, the rulesof vector addition, subtraction and displacement in the color spacerepresent the CNS or ANS-space. Logic operations then become the rulesof colorimetry.

For example, let p be red (R), say 620 nm., p a complementary cyan (C)at 494 nm., q a yellow (Y) at 575 nm., and q a blue (B), complementaryto Y, at 470 nm. Then, working in the CNS-space with these hueassignments, we can compute modus ponens colorimetrically, for exampleas illustrated in FIG. 48. The addition of p or R, for example 620 nm.,and −p v q or CY, for example having a dominant wavelength around 530nm., with colorimeter 4802 is the yellow Y of 575 nm.: q. The“tautology” at white (W) is the addition of complementary hues.

In the ANS-space black is the contradiction. With complementaries in theANS-space’ . . . what is offered, so to speak, by one [reflection]spectrum (or colour) is withdrawn by the other, so that the result is avanishing of colour, just as in the contradiction between twopropositions which negate one another the result is a vanishing ofinformation’ (Jonathan Westphal, Colour, Oxford, Blackwell, 2nd. ed.,1991, p. 108).

In the ANS-space we can perform exactly parallel computationscolorimetrically, in particular simplification routines. Take the mostbasic simplification as an example as illustrated in FIG. 49, whereinthe colorimetric cell will yield a signal of a specified output, say forpq v −p−q. Let p be R, for example 620 nm., let q be Y, for example 575nm., let −p be C, for example 494 nm., and let −q be B, for example 485nm. The input p is reflected off mirror 4902 to be combined with theinput q with combiner 4904. The input −q is reflected off mirror 4910 tobe combined with the input −p with combiner 4912. The combination pq isreflected off mirror 4906 to be combined with the combination −p−q withcombiner 4908. The calorimeter 4914 provides an output of p, or in otherwords R, from input pq v −p−q. The analog of the sigmoid filter in thisparticular colorimetric application of vector logic is the color filterwhose ideal transmission curves are shown in FIG. 50, with90%+transmission peaks at 485 nm. and 595 nm. It is an OR-gate; there isa signal through the filter iff either p and q or −p and −q. As such,the schema pq v −p−q is logically equivalent to p. YR will represent pq,and BR is pq. The Y and B components cancel, leaving RR or R, which isp.

(v) Polarization Based Processing

The vector logic system also provides for optical AND, NAND, NOT, andthe other logical functions, implemented in optical gates in which theinput and output are coded directionally in a more purely geometricalform. The implementation does not call for a nonlinear optical material,but instead embodies the vector logical analysis of the AND-function inits developed CNS-form, and well-known optical materials: reflectors ormirrors, beamsplitters, retarders and polarizers.

Each gate, or cell, may comprise of a group of optical elements arrangedin three layers. The elements are reflectors, polarizers and retarders.The three layers in each cell represent the conjunctional normal form(CNS-plane), the alternational normal form (ANS), and an intermediatetransformational layer (T-plane). The input to each cell is the opticalbeam (or beams) which enter the cell, from any input direction. Theoutput from each cell is the optical beam (or beams) which leave thecell.

FIG. 51 illustrates an exemplary AND-cell 5100 in accordance with oneembodiment of the present invention. Consider the input of p 5108 and q5110 into the top CNS-plane 5102. A p v q input would occur only on theANS-plane. The two entry vectors, p and q travel towards the center ororigin O-CNS of the cell. At O the p-beam 5108 is split by beam splitter5112 into two optical components, traveling to the points p v q and p v−q. This is the optical analog of the developed form of p. Pairs ofreflectors and polarizers are placed at the corners of the cell, as wellas a 180 mμ phase-shifter or retarder in two of the adjacent corners.The q-beam is also split into its developed form by the beam-splittersat O, and directed to −p v q and to p v q.

At this point the three resultant beams (−p v q, p v q, and p v −q) aredirected to the center of the T-plane. When the two polarized beams −p vq and p v −q meet at the origin 0₁T they are extinguished, as they areout of phase by 180°. If the distances from 0₁T to the vertices of thecell are correctly set, then there will be a local null-value for theoutput at the points at which the exit vectors emerge from the cell,i.e. at p v −q and −p v q, as well as at 0 ₁T. The T-plane also has asecond sub-plane on which the p v q and −p v −q beams would meet, with acenter at 0 ₂T. But in the case of the three beams from p and q inputs,there is no −p v −q beam, and the p v q beam, unextinguished, travelsonto the ANS-plane. It continues as the exit vector or output of theANS-plane in a pq direction: it has become the output pq. At this pointthe output pq can be reflected as new input to the next cell, and thefurther outputs arranged in a cascade of cells with the different logicfunctions. But if the output is negative, it must be entered asnegative, −p or −q.

If for example a combination of inputs is given, which is part of −(pq),i.e. any of −pq or p−q or −p−q, then the exit vectors will be one ormore of these conjunctions on the ANS-plane, and this will be routed asnegative output. When such a negative output is fed into a NOT-cell5204, which has the function of inverting all inputs by reflection androuting them to the dual plane, it is output as a positive signal, asillustrated in FIG. 52.

Specifically, when the output of the AND-cell 5202 is input in to theNOT-cell 5204, the final output is −p if pq is input. If p and q areinput, −(pq) is “logically” off though optically on as the −p exitvector and p entry vector in the next cell. It is an important featureboth of the vector logic system itself given in “Logic as a VectorSystem” and of the present directional implementation that a negativelogic output, −p, say, is optically positive, just as −p is vectoriallypositive in the sense that there is a directed line segment −p, whichhappens to point in a reverse p direction. If −p is true then the −pbeam is on, though logically negative, as all left-hand or down-tendingbeams in the CNS-plane are coded negative. An important related featureof the implementation is that input beams are to be set in defaultnegative states such as −p, −q, etc., for the first-level inputs, sothat at the onset of computation the inputs to the system are logicallyoff though optically on.

Consider the implementation of the COIN-function or coincidencefunction, in the cells shown in FIG. 53. The initial off-inputs are −pand −q in both AND-cells 5302 and 5304. Without on-input, the −p−q-cell5302 gives an output to the OR-cell, while the pq-cell 5304 does not.What happens when the overall input is say p−q? In both the AND-cells at5302 and 5304, there is no positive output, and hence no positive outputin this case for the pq v −p−q cell. The output in the pq-cell isnegative, and hence fed to −p in the OR-cell 5306. So the OR-cell 5306registers a NOT-state if p−q is input into the AND-cells 5302 and 5304.Only at the final output or outputs should the values of the logic andoptical positive and negative be caused to coincide, and negativesignals be extinguished.

Precisely the same principles apply in the implementation of theXOR-function, or half-adder FIG. 54 and to the MUX-function ormultiplexer FIG. 55.

Finally, to illustrate how cascades of more than one level can beimplemented in the optical logic described here, consider how theNAND-function provides a way of saying pq. The following is equivalentto pq: (pNANDq)NAND(pNANDq). What we have to do to create an opticalcircuit which expresses this function is to build a sequence of twoNAND-cells conjoined into another NAND-cell. FIG. 56 shows this sequencefor an input of p and q, FIG. 57 for an input of p and −q. This examplealso shows, of course, that any logical function and any opticalintegrated circuit can be implemented within the optical logic which hasbeen described.

It is worth emphasizing again that these gates and circuits in thefourth embodiment use only four principles of implementation:beamsplitting; reflection; polarization; and retardation. These areimplementations which can be scaled down as far as desirable. There isno scale at which reflection of a stream of light cannot take place, andeven at the smallest level photons can be polarized simply by areflecting surface, as, for example, on a highly polished metallicsurface of an automobile. As for retardation, there is a sense in whichit does not even need a physical embodiment within the system, as it canbe organized by the architecture of the gates involving different pathlengths of the corresponding beams.

FIG. 58 depicts an exemplary embodiment of a NOR-gate, orSheffer-function −(p v q), in accordance with the present invention. Inthe cell 5802 at the origin, four beam-splitters 5820 break up inputfrom p, q, p and q into vectors exiting as the developed form, so thatp, for example, becomes p q v. pq. The exit vectors from the cell arethe normal of the (p, q) point coordinates, for example exits at p q.

Exit vector e point positions: p q 7:30 pq 4:30 pq 11:30  pq 1:30

The exit vector for {right arrow over (pq)} is blocked in the nor-cell,represented by the shaded port at pq.

(Two orthogonally oriented polarizers are present at the entry to theports pq and p q

Suppose now inputs in the following pairs of literals:

a) pq b) p q c) pq d) p q

These inputs will explain further features of the cell as well as itsoverall “

” function and output. For example as illustrated in FIG. 59.

The input beams p and q are split at O to p 1 v pq and pq v pqrespectively. This gives a double beam to p 1, which is Π 1. But the p qv pq output port is blocked. At the p q and pq outputs the two beams areorthogonally polarized, and they are then directed together at E_(x),the xnor sum at the top right of the cell. (The actual spatial location)of these last ports E_(x), and E_(f) is arbitrary.). As there is nooutput at pq, the final output, here labelled E_(f), is zero, whereinthere is no light, and (p:1, q:1=0).

-   -   b) p q—Take next the input (p, 1), as in FIG. 60.

The input beams p and q are split at O to pq v. p q→and p q vpq→respectively. The pq→ output is blocked as before, but it is only 25%of the total output, which is itself twice the input. The p q→output ispolarized at zero°, but there is no p q→output, so that E_(x) is on, andsince pq→is on, so is E_(f), consequently (p=1, 1:1,=1). So far we have:

pq pq; 1 1 0 1 0 c) 1 1q

In FIG. 61, the input is ( pq). The input beams p q→are split at O togive developed pq v. pq→and pq v. pq→respectively. The double pq beam ispolarized and reflected to E_(x) and thence to E_(f), as there is noorthogonally polarized p q→output at pq. The pq→output is blocked, asbefore, but E_(f)=1, even without the contribution of p q→

We now have for the cell at E_(f):

pq pq; 1 1 0 1 0 1 0 1 1

-   -   d) Turning finally to input ( pq), as in FIG. 62, the logical        pattern is similar. The E_(x) or XNOR-function is extinguished,        but the double p q→component brings the sum to 1 at E_(f). The        block at ( pq) is unnecessary as there is no pq→components at        this exit port.

The result overall is p

q for the cell is:

pq pq 11 0 10 1 01 1 00 1

Furthermore, exploiting the geometrical nature of the light beams in amatrix around the origin rather than in an optically bistable physicalmedium, is conceived as functioning like a silicon plate in asemiconductor chip.

The basic cell concept here described is complete, and can by itself beused to construct cascades of other functions in a gate sequence. But inaddition, we can exploit the concepts to devise any other logicfunction, most importantly p

q the NAND or Pierce function in the dual CNS-space or vector sub-space.These cells are all-optical and can be put together to implement thelogic functions of decoders, multiplexers, adders, and the rest,including of course, router switches.

(vi) Polarization Based Processing with a Dualization/PolarizationCorridor There is another way to implement the vector logic system usingpolarization which does away with the potential traffic jam at thecenter of the AND-cell. Let us take once again AND-function, andconsider two inputs, p and q, which enter the AND-cell shown in FIG. 63.

At each literal (p, −p, q, −q) of the input plane cell there is abeam-splitter will have the effect of dividing the p and q beams intotwo beams each. These two beams will represent p v q and p v−q, for p,and p v q and −p v q, for q. The logical equivalent of “p and q” usingthe “or” function is the so-called developed conjunctional normal form“(p v q) and (p v −q) and (−p v q)”, as can be seen from the truth-tablein FIG. 64. What has happened is that optically, using the vector logicrepresentation, we have modeled this form. The same would happen withother literal inputs −p and −q.

The next step is to realize something interesting and important aboutthe developed conjunctional normal form. If we take the first conjunct,in this case p v q, we can arrive at the proposition “p and q” bychanging the “or” to an “and”, and dropping the other two conjuncts. Butwhat we can notice here about the other two conjuncts is that they areopposed vectors in the vector logic space: they are p v −q and −p v q.This rule applies no matter how large the conjunction is. Say it's −pand q and r”, an 8-line table. Then we can take the rather largedeveloped conjunctional normal form, note that the conjunction has T onevery line of the truth-table except the last, and that the remaining 2to the Nth minus 2 rows (6 of them) are opposed in vector space.Examining the truth-table in Diagram 2, we can see that any conjunctionof no matter what length can be expressed by deleting the vector opposed(or technically S-opposed) rows, in this case Rows 2 and 3), andconverting the main operator (or) into its dual. It is as if one hadsaid that if the S-opposed values are deleted, then the derived valuesfor pq and p v q are the same, and p v q is pq!

An optical analog of this is a polarization cancellation of one beam byanother. Imagine that the three beam groups we have in the cell on theCNS-plane ((p v q)(p v −q)(−p v q)) are now reflected and, at theentrance to the polarization corridor, polarized orthogonally withrespect to one another, in such a way that the opposed vectors in theCNS-plane cancel, for example p v q and −p v −q, or p v −q and −p v q.We have the extinction at a point at which the two “logically opposed”beams are incident. We can tag the exit vector at 1:30 in the exit planein Diagram 1 pg, and we can enter it into other cells as an input, p orq or whatever. Thus we have a cascade, as before in (iv), and thegeneral structures of these cascaded cells are the same.

FIG. 63 also shows how the beams in the dualization or polarizationcorridor are to be managed to as to drop Operator S-opposed disjuncts ofconjunctions. (Conjunctions of disjunctions with opposite literal valuesare Operator S-opposed, disjunctions of conjunction are OperatorN-opposed.). When p and q are entered on the input plane, in the lowerfour lines of polarization corridor (representing the XOR-function) weget p from p v −q and −p from −p v q. But these are orthogonallypolarized and cancel. In the upper four lines of the polarizationcorridor representing pq and −p−q, the so-called COIN-function for(COINcidence), the p and q beams have the same plane of polarization andso do not cancel one another. The polarizations governing theXOR-function in the bottom for lines of the dualization corridor ensurethat the extraneous non-AND elements of the conjunctive normal formexpansion for any coincident conjunction are extinguished, and theCOINfunction dos the same for the non-coincident conjunctions such as pand −q.

FIG. 65 shows an all-optical data distributor, decoder or DMUX (logicaldemultiplexer), with a message r being delivered to Alice, at addresspq, or to Bob, at p−q.

FIG. 66 shows a latching function for a simple set-reset flip-flop forall the possible optical paths. Here it is important to note that evenwhen the unit is logically off, optically it is on, and this is how thestored “set” value Q is held. In this way optical storage is achievedusing the AND-function within the NOR-function version of the SRflip-flop.

FIG. 67 shows the same flip-flop for values S=1, R=O, present Q=1, for anew Q or output of 1.

In conclusion, consider four interesting features of the optical system.Interesting Feature A is that in the vector logic system there is asense in which reverse vectors are both positive. In the implementationwe see that both p and −p for example are equally real optically. Sologic negative is not optical negative. Of course at the end of thedesired computation, the SR flip-flop-function for example, we takeaccount of the positive output, and ignore the negative output. At thispoint optical positive and logic positive will coincide. This feature isimportant for the flip-flop, as the light for the latched value comesfrom the optically positive signal even when that value is negative.

Interesting Feature B is connected with this. It is that when data isentered into the system, say p and −q, then that will be a matter ofkilling off the −p beam and killing off the q-beam. We must imagine thatinitially all the beams are optically on, and that we select our inputby removing the beam in the opposite direction. Or if you like we canimagine that at the outset the default state is one in which the −p beamand the −q beam are on, and that they stay on unless redirected.

Interesting Feature C is that for any function built out of the AND andthe NAND-functions if the input power of the variables such as p and qis 1, then the output is 1, leaving aside operating losses fromimperfect polarization extinction ratios, losses on reflection and soforth.

Interesting Feature D is perhaps the most important. The polarizationbased-implementations are clearly reversible, in the sense that noinformation is destroyed anywhere in the system. What the vector logicimplementation does is to move information around, without destroyingany physically, and to selectively use the output. In the case of aNAND-cell, for example, the negative outputs are fed as parallel beamsinto the positive input of the next cell. They are as close together asis necessary for the next stage of processing, but they are distinctbeams. Any input into a cascade is recoverable from the output, andinformation is conserved.

Can the implementation of the present system be reconciled with thecharacter of the information in the bitstream emerging from a fiberoptic cable? That is, is an OEO-conversion required before informationin the bitstream can undergo logical processing with the vector logicsystem in the implementations described in (iv) and here in (v)? Theanswer is that the mode of the signal of the fiber can be made to encodespatial information, at least over relatively short distances, evenafter enormous twisting and even knotting of the cable. An initialcalibration can be used to determine the exit vectors of the p and qvariables at the end of the fiber.

FIG. 68 shows the coding of p, q, −p and −q in the propagation of lightrays through a short piece of fiber. For the correct length the inputwill match the output. So the with this method of coding the spatiallyor vector coded information can be introduced directly into a processingunit composed of the optical AND, NAND and other gates described.

FIG. 69 illustrates an optical AND gate and switching device usingpolarization photochromism. The AND-gate described in FIG. 69 is aswitch having an optical output in a selected direction of 1 iff (if andonly if) the inputs from the incident beams, here called the p- andq-beams, are both 1. The pq-beam (p and q) is on iff the p-beam is onand the q-beam is on. This is achieved by using: a polarizationphotochromic with a given plane of polarization as the so-called“switching medium”; a q-beam has a control signal which shifts thepolarization plane of the switching medium ninety degrees; and a p-beamwhose plane of polarization is orthogonal to the original plane ofpolarization of the switching medium. When the q-beam is off, the p-beamsuffers extinction in the switching medium. But the q-beam is directedat an angle which is not the output direction. The result is a genuineoptical AND-gate which can be used in switching and processingapplications.

The AND-function is basic for the other more complex logical functionsnecessary in high-speed photonic signal systems in telecommunicationsand computational applications generally. This has been the source ofthe great interest taken recently in optically bistable devices andmaterials. Yet optical materials seem intrinsically unsuited to theproduction of bistable state, unlike their analogs in semiconductortechnology. However, the polarization photochromics are an exception, asthey are materials which do exhibit the demand characteristics, in thegiven configuration. These characteristics are given in the truth-table:

pq pq 11 1 10 0 01 0 00 0

More complex optical switches such as multiplexers and demultiplexers,which are themselves logical functions of input or switches cantherefore be constructed. It is commonly recognized that the changes inthe absorption profiles of photochromic polymers generally haveoptoelectronic, optical storage and logic-switching applications, butthe uniqueness of the present invention is that it shows how the lattercan be made specifically with the polarizing photochromics.

The heart of the optical AND-gate or switch described here (FIG. 1),which is entirely novel and for which there is no prior art, is aswitching medium fabricated of a material such as but not limited to theliquid crystalline line polymers listed by Ichimura, described in thebackground of the invention, above, which display the requiredphotochromism with respect to polarization

In FIG. 69, the p-beam is directed towards the output through theswitching medium, and is in a polarization state orthogonal to that ofthe polymer. It therefore suffers extinction at the surface of thepolymer unit.

The q-beam acts as an angle to the p-beam. It consists of polarizedactinic or UV light. It therefore induces a 90-degree photochromicchange in the angle of polarization or polarization sate of theswitching medium. The q-beam is arranged to strike the polymer cellbefore the p-beam e.g. lengthening the path of the q-beam relative tothe p-beam. It reorients the plane of polarization of the polymer unitprior to the arrival of the p-beam. The result is that the unit nowblocks the p-beam.

It is still true, as Norbert Streibl et. al. (“Digital Optics”, (1989))pointed out, the ‘A uniform technology for digital optical informationprocessing, comparable in its significance to microelectronics, does notyet exist and is by itself a challenging research goal.’ A vector logicfor optics is a source from which such a “uniform” technology can flow,just as electronics derived from the natural isomorphism of electriccircuitry and truth-functional logic.

In this disclosure, there is shown and described only the preferredembodiment of the invention, but, as aforementioned, it is to beunderstood that the invention is capable of use in various othercombinations and environments and is capable of changes or modificationswithin the scope of the inventive concept as expressed herein.

Although certain specific embodiments of the present invention have beendisclosed, it is noted that the present invention may be embodied inother forms without departing from the spirit or essentialcharacteristics thereof. The present embodiments are therefore to beconsidered in all respects as illustrative and not restrictive, thescope of the invention being indicated by the appended claims, and allchanges that come within the meaning and range of equivalency of theclaims are therefore intended to be embraced therein.

1. An optical NAND-gate comprising: first, second, third, fourth, fifth,and sixth optical inputs; first, second, third, fourth, fifth, and sixthoptical outputs; first and second optical paths leading from said firstand second respective optical inputs to said first optical output; thirdand fourth optical paths leading from said third and fourth respectiveoptical inputs to said second optical output; fifth and sixth opticalpaths leading from said fifth and sixth respective optical inputs tosaid third optical output; intensity filters located within each of saidfirst, second, third, fourth, fifth, and sixth optical paths; a firstoptical filter having a sigmoid characteristic located at a positioncommon to both said first and said second optical paths; a secondoptical filter having a sigmoid characteristic located at a positioncommon to both said third and said fourth optical paths; and a thirdoptical filter having a sigmoid characteristic located at a positioncommon to both said fifth and said sixth optical paths; wherein saidfourth, fifth, and sixth optical outputs from said respective first,second, and third optical filters having a sigmoid characteristic beingin communication with a fourth optical filter having a sigmoidcharacteristic.